From (REF ) we have
\(\left| G(-t) \right| \le (1-\alpha )^{-\gamma } \frac{\Gamma (1-\Re (t))}{\left|\Gamma (1-t)\right|} ,\) 
and from Eq. 2.4.4 of §2.4 of [1]} we have
\(\log \left| \Gamma (t) \right| \mathop {\rightarrow }\limits _{\left| \Im (t) \right| \rightarrow \infty } & -\tfrac{1}{2}\pi \left| \Im (t) \right| + (\Re (t)-\tfrac{1}{2})\log \left| \Im (t) \right| ,\\\log \left| \Gamma (1-t) \right| \mathop {\rightarrow }\limits _{\left| \Im (t) \right| \rightarrow \infty } & -\tfrac{1}{2}\pi \left| \Im (t) \right| + (\tfrac{1}{2}-\Re (t))\log \left| \Im (t) \right| .\) 

Thus the exponential decay of the integrand vanishes and one is left with the algebraic decay of (REF ).
